Horizontal Curves by Jon B Purnell PLS 2008

Parts of a Circle • HORIZONTAL CURVES are based on circles • CIRCLE: set of points equidistant from a center point - (RP) • Radius - (R) distance from the RP to any point on the circle • Arc any subset of points on the circle

Arc Length • CURVES are portions of circles: they are circular arcs • If circumference = 360º, = (2* *R) = 1570. 80 ft… • …what is length of the Arc? • Arc = (90º/360º) of circumference, or 25% of circle. . . • …so Arc = (2* *R)*( / 360º) = 392. 70 ft.

Parts of a Curve Point of Intersection Point of Tangency Point of Curvature Radius Point

Geometry: Perpendiculars Perpendicular

Geometry: Delta Angle Delta ( ) ( /2) Delta ( )

Parts of a Curve Long Chord (C)

Parts of a Curve Arc Length (L)

Parts of a Curve Tangent (T)

Parts of a Curve External (E)

Parts of a Curve Middle Ordinate (M)

Computing Arc Length =90º R=250 L=2 R( /360º) L=2 250(90º/360º) =392. 70

Key to the Chord: triangle PC -PT-RP =90º R=250 sin( /2)=(. 5 C/R) 0. 70711=. 5 C/250 0. 70711*250=. 5 C C=353. 55

Key to the Tangent: triangle PC -PI-RP =90º R=250 tan( /2)=T/R 1=T/250 T= 1*250. 00

Keys to the External: Radius and triangle PCPI-RP =90º R=250 E=(R/cos( /2))-R C=(250/cos(45º))-250 =103. 55

Keys to the Middle Ordinate: Radius and triangle PC-PTRP =90º R=250 M=R-(R*cos( /2)) M=250 -(250*cos(45º)) =73. 22

Stationing PI Sta. 4+10. 23 PI Sta = PC Sta + T PT PC + a St PC 52 a = 5+ PT St St a. 3 . 2 60 . 9 3 1+ a. L St Begin Project Sta 0+00

• required to subtend a 100 foot arc • Smaller D means larger R • Simplifies layout • R = 5729. 578 / D • D = 5729. 578 / R Degree of Curvature: Arc Definition

Degree of Curvature • Simplifies layout • Defl. angle to POC = • D*n/2 • D = degree of curve • n = stations • 12º *3/2 = 18º

Curve Staking I Defl < to POC 5+00 = D*n/2 D = degree of curve n = stations 12º *2. 55/2 = 15º 18’ Chord to POC 5+00 = 2*R*Sin(15º 18’) = = 251. 98 D=12º R=477. 46

Curve Staking II • TD = R * sin (Delta POC) • TO = R - R * cos(Delta POC)

Compound Curves • Two curves back to back • RP’s on same side of centerline • Compute like two simple curves

Reverse Curves • Two curves back to back • RP’s on opposite sides of centerline • Compute like two simple curves

Circular Areas Circular sector • Area of a circle = p. R 2 • Area of Circular sector = (p. R 2) *(D/360) Annulus: area between concentric circles • Area of Annulus = p. R 12 -R 22 Annular sector • Area of Annular sector = sector 1 - sector 2 • = (p. R 12 -R 22)*(D/360)